Derivative pricing using neural networks#
Over the past few years, advancements in artificial intelligence and machine learning have naturally led to the adoption of these techniques in many other fields. Neural networks, for example, have become a highly flexible tool capable of solving a wide variety of problems such as image classification, speech recognition, and language translation.
Today, we also see their application in various industries, including finance. In this notebook, we demonstrate how these models can be used—for instance, to price derivatives. Although derivative pricing is a relatively simple and well-studied problem, it provides an interesting example to illustrate how neural networks can solve financial problems, not only in pricing but also in inverse problems (e.g., model calibration) and risk management.
In this entry, we focus on a particular technique: Physics-Informed Neural Networks (PINNs). As the name suggests, these neural networks are designed to incorporate additional information derived from physical laws or known mathematical relationships. Typically, this is achieved by embedding these laws into the loss function to guide the network toward a solution that satisfies the given differential equation.
What are Physics-Informed Neural Networks (PINNs)?#
First we begin with some definitions. Suppose we wish to solve a partial differential equation (PDE) of the form \( \mathcal{N}[V(\mathbf{x})] = 0,\quad \mathbf{x} \in \Omega, \) where:
\(\mathcal{N}\) is a differential operator,
\(V(\mathbf{x})\) is the unknown solution,
\(\Omega\) is the spatial (and possibly temporal) domain.
In a PINN, we approximate \(V(\mathbf{x})\) with a neural network \(V_\theta(\mathbf{x})\) parameterized by \(\theta\). The loss function is constructed to penalize deviations from both the governing PDE in the domain and the prescribed boundary conditions on \(\partial \Omega\). Formally, the loss can be written as:
with
and
where:
\(\{\mathbf{x}_r^i\}_{i=1}^{N_r}\) are the collocation points in the interior of the domain \(\Omega\),
\(\{\mathbf{x}_b^j\}_{j=1}^{N_b}\) are the points on the boundary \(\partial \Omega\) with known values \(g(\mathbf{x}_b^j)\).
In order to train the network, we minimize the loss function using gradient-based optimization techniques. The resulting neural network will approximate the solution to the PDE in the domain \(\Omega\) and satisfy the boundary conditions on \(\partial \Omega\).
The idea is not to overcomplicate things, so we start with the example. First, we import the necessary libraries:
Show code cell source
import torch
import torch.nn as nn
import torch.optim as optim
import numpy as np
from torch.autograd import grad
from scipy.stats import norm
device = torch.device("cuda:0" if torch.cuda.is_available() else "cpu")
np.random.seed(42)
torch.manual_seed(42)
A module that was compiled using NumPy 1.x cannot be run in
NumPy 2.2.3 as it may crash. To support both 1.x and 2.x
versions of NumPy, modules must be compiled with NumPy 2.0.
Some module may need to rebuild instead e.g. with 'pybind11>=2.12'.
If you are a user of the module, the easiest solution will be to
downgrade to 'numpy<2' or try to upgrade the affected module.
We expect that some modules will need time to support NumPy 2.
Traceback (most recent call last): File "<frozen runpy>", line 198, in _run_module_as_main
File "<frozen runpy>", line 88, in _run_code
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel_launcher.py", line 18, in <module>
app.launch_new_instance()
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/traitlets/config/application.py", line 1075, in launch_instance
app.start()
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/kernelapp.py", line 739, in start
self.io_loop.start()
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/tornado/platform/asyncio.py", line 205, in start
self.asyncio_loop.run_forever()
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/asyncio/base_events.py", line 608, in run_forever
self._run_once()
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/asyncio/base_events.py", line 1936, in _run_once
handle._run()
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/asyncio/events.py", line 84, in _run
self._context.run(self._callback, *self._args)
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/kernelbase.py", line 545, in dispatch_queue
await self.process_one()
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/kernelbase.py", line 534, in process_one
await dispatch(*args)
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/kernelbase.py", line 437, in dispatch_shell
await result
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/ipkernel.py", line 362, in execute_request
await super().execute_request(stream, ident, parent)
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/kernelbase.py", line 778, in execute_request
reply_content = await reply_content
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/ipkernel.py", line 449, in do_execute
res = shell.run_cell(
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/ipykernel/zmqshell.py", line 549, in run_cell
return super().run_cell(*args, **kwargs)
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/IPython/core/interactiveshell.py", line 3077, in run_cell
result = self._run_cell(
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/IPython/core/interactiveshell.py", line 3132, in _run_cell
result = runner(coro)
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/IPython/core/async_helpers.py", line 128, in _pseudo_sync_runner
coro.send(None)
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/IPython/core/interactiveshell.py", line 3336, in run_cell_async
has_raised = await self.run_ast_nodes(code_ast.body, cell_name,
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/IPython/core/interactiveshell.py", line 3519, in run_ast_nodes
if await self.run_code(code, result, async_=asy):
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/IPython/core/interactiveshell.py", line 3579, in run_code
exec(code_obj, self.user_global_ns, self.user_ns)
File "/var/folders/cp/l432p0ns1t38qmnyr_3l3r000000gn/T/ipykernel_71774/4220564520.py", line 1, in <module>
import torch
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/__init__.py", line 1477, in <module>
from .functional import * # noqa: F403
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/functional.py", line 9, in <module>
import torch.nn.functional as F
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/nn/__init__.py", line 1, in <module>
from .modules import * # noqa: F403
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/nn/modules/__init__.py", line 35, in <module>
from .transformer import TransformerEncoder, TransformerDecoder, \
File "/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/nn/modules/transformer.py", line 20, in <module>
device: torch.device = torch.device(torch._C._get_default_device()), # torch.device('cpu'),
/Users/josemelo/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/nn/modules/transformer.py:20: UserWarning: Failed to initialize NumPy: _ARRAY_API not found (Triggered internally at /Users/runner/work/pytorch/pytorch/pytorch/torch/csrc/utils/tensor_numpy.cpp:84.)
device: torch.device = torch.device(torch._C._get_default_device()), # torch.device('cpu'),
<torch._C.Generator at 0x117983790>
Loss functions and the governing equation#
For the problem of pricing options, the fundamental PDE we need to solve is the Black-Scholes equation. This equation is a parabolic PDE that describes the evolution of the price \(V(S,t)\) of an option in terms of time \(t\) and the underlying asset price \(S\). It is given by:
where:
\(V(S,t)\) is the option price,
\(S\) is the price of the underlying asset,
\(r\) is the risk-free interest rate,
\(\sigma\) is the volatility of the underlying asset.
Our objective is to use this PDE to guide the training of our neural network. Modern machine learning libraries, such as PyTorch, offer automatic differentiation capabilities. For example, torch.autograd.grad efficiently computes the necessary derivatives, such as \(\nabla V(S,t)\), at each training point. In our fomulation, \(\mathcal{L}_{\mathrm{PDE}}(\theta)\) would be represented by the following function:
def pde_loss_f(model, inputs, sigma, r):
inputs.requires_grad_(True)
V = model(inputs)
# First order
gradients = grad(V, inputs, grad_outputs=torch.ones_like(
V), create_graph=True)[0]
dVdt = gradients[:, 0]
dVdS = gradients[:, 1]
# Second order
d2VdS2 = grad(dVdS, inputs, grad_outputs=torch.ones_like(
dVdS), create_graph=True)[0][:, 1]
S = inputs[:, 1]
V = V[:, 0]
# PDE
pde_residual = dVdt + 0.5 * sigma ** 2 * S ** 2 * d2VdS2 + r * S * dVdS - r * V
loss_pde = torch.mean(pde_residual ** 2)
return loss_pde
Additionally, to incorporate the boundary conditions, we need to define an additional loss functions that force the network to satisfy these constraints. Since the boundary conditions we are using are of Dirichlet type, the loss function is relatively simple (it merely compares the network outputs against the ground truth). However, nothing prevents us from employing other types of boundary conditions (such as Neumann or Robin).
def boundary_loss_f(model, inputs, outputs):
V_pred = model(inputs)
boundary_loss = torch.mean((V_pred - outputs) ** 2)
return boundary_loss
Collocation points#
To solve the PDE using PINNs, we need to define the numerical domain over which the equation will be solved. Much like traditional numerical methods (e.g., finite differences), we select a set of points -called collocation points- where the PDE is enforced. These include points in the interior of the domain \(\Omega\) as well as on its boundary \(\partial\Omega\).
To enforce the boundary conditions, additional loss terms are added that force the network to satisfy them. For a European call option, the typical boundary conditions are:
At \(S = 0\): $\( V(0, t) = 0, \)$ since if the underlying asset’s price is zero, the option is worthless.
For \(S \to \infty\): $\( V(S, t) \approx S - K e^{-r(T-t)}, \)\( as the option behaves like a linear payoff for large \)S$.
At expiration \(t = T\): $\( V(S, T) = \max(S - K,\, 0), \)$ which is simply the payoff of the option.
We sample this points using random numbers, and we will use them to train the neural network. Next is an implementation for sampling this points:
def payoff(s, k):
return np.maximum(s-k, 0)
def interior_samples(option_config, scaling=4):
t = np.random.uniform(
0, option_config['maturity'], option_config['n_samples'])
s = np.random.uniform(
0, scaling*option_config['strike'], option_config['n_samples'])
tag = np.zeros(option_config['n_samples'])
output = np.zeros(option_config['n_samples'])
return t, s, tag, output
def top_boundary_samples(option_config, scaling=4):
t = np.random.uniform(
0, option_config['maturity'], option_config['n_samples'])
s = np.ones(option_config['n_samples']) * scaling*option_config['strike']
strike = option_config['strike']
s_max = scaling*option_config['strike']
r = option_config['r']
T = option_config['maturity']
output = s_max - strike * np.exp(-r*(T-t))
tag = np.ones(option_config['n_samples'])
return t, s, tag, output
def bottom_boundary_samples(option_config):
t = np.random.uniform(
0, option_config['maturity'], option_config['n_samples'])
s = np.zeros(option_config['n_samples'])
output = np.zeros(option_config['n_samples'])
tag = np.ones(option_config['n_samples'])
return t, s, tag, output
def initial_condition_samples(option_config, scaling=4):
t = np.ones(option_config['n_samples']) * option_config['maturity']
s = np.random.uniform(
0, scaling*option_config['strike'], option_config['n_samples'])
tag = np.ones(option_config['n_samples'])
output = payoff(s, option_config['strike'])
return t, s, tag, output
Important
The choice of collocation points is critical for the convergence of the method. One potential improvement is to use quasi-random numbers to distribute the collocation points more uniformly across the domain, thereby avoiding clustering that can occur with some random number generators.
Of course, we need to define the financial parameters that will be used in the problem. For this example, we will use the following values:
config = {'strike': 15, 'maturity': 1,
'r': 0.04, 'sigma': 0.25, 'n_samples': 10000}
# Generate samples
inner_samples = interior_samples(config)
top_samples = top_boundary_samples(config)
bottom_samples = bottom_boundary_samples(config)
initial_samples = initial_condition_samples(config)
Visually, this is how our domain looks like:
The neural network#
In this exercise, we employ a relatively simple feedforward neural network with one hidden layer, a few neurons, and \(\tanh\) activation functions. While more complex architectures may be required for higher-dimensional problems, empirical evidence suggests that a simple network is sufficient for this particular case.
class PINN(nn.Module):
def __init__(self):
super(PINN, self).__init__()
size = 15
self.hidden_layers = nn.ModuleList([
nn.Linear(2, size),
nn.Linear(size, size),
])
self.output_layer = nn.Linear(size, 1)
def forward(self, inputs):
x = inputs
for layer in self.hidden_layers:
x = torch.tanh(layer(x))
x = self.output_layer(x)
return x
It is also important to initialize the network weights properly, as this can affect the convergence of the solution. Unlike typical machine learning tasks that require large amounts of data, PINNs can often solve PDEs with relatively little (even synthetically generated) data. Therefore, global optimizers such as L-BFGS, which use the entire dataset when updating the weights, are particularly effective.
def initialize_weights(model):
for layer in model.hidden_layers:
if isinstance(layer, nn.Linear):
nn.init.xavier_uniform_(layer.weight)
nn.init.zeros_(layer.bias)
Training loop#
Now with all set, we can proceed to train the neural network.
def train_model(model, inputs, outputs, tags, sigma, r, iters=1000):
optimizer = optim.LBFGS(
model.parameters(),
max_iter=iters,
lr=1,
line_search_fn='strong_wolfe',
tolerance_grad=1e-10,
tolerance_change=1e-10
)
loss_history = {'total_loss': []}
def closure():
optimizer.zero_grad()
total_loss = 0
pde_mask = (tags == 0).squeeze()
boundary_mask = (tags == 1).squeeze()
if pde_mask.any():
pde_loss = pde_loss_f(model, inputs[pde_mask], sigma, r)
total_loss += pde_loss
if boundary_mask.any():
boundary_loss = boundary_loss_f(
model, inputs[boundary_mask], outputs[boundary_mask])
total_loss += boundary_loss
print(
f'PDE Loss: {pde_loss.item()} | Boundary Loss: {boundary_loss.item()}', end='\r', flush=True)
total_loss.backward()
loss_history['total_loss'].append(total_loss.item())
return total_loss
optimizer.step(closure)
return loss_history
Show code cell source
# input samples
inner_s = torch.stack([torch.tensor(x, dtype=torch.float32)
for x in inner_samples[0:2]], dim=0).T
top_s = torch.stack([torch.tensor(x, dtype=torch.float32)
for x in top_samples[0:2]], dim=0).T
bottom_s = torch.stack([torch.tensor(x, dtype=torch.float32)
for x in bottom_samples[0:2]], dim=0).T
initial_s = torch.stack([torch.tensor(x, dtype=torch.float32)
for x in initial_samples[0:2]], dim=0).T
# output
inner_output = torch.tensor(inner_samples[3], dtype=torch.float32)
top_output = torch.tensor(top_samples[3], dtype=torch.float32)
bottom_output = torch.tensor(bottom_samples[3], dtype=torch.float32)
initial_output = torch.tensor(initial_samples[3], dtype=torch.float32)
# tags
inner_tags = torch.tensor(inner_samples[2], dtype=torch.float32)
top_tags = torch.tensor(top_samples[2], dtype=torch.float32)
bottom_tags = torch.tensor(bottom_samples[2], dtype=torch.float32)
initial_tags = torch.tensor(initial_samples[2], dtype=torch.float32)
# concatenate
inputs = torch.cat([inner_s, top_s, bottom_s, initial_s], dim=0)
outputs = torch.cat([inner_output, top_output, bottom_output,
initial_output], dim=0).reshape(-1, 1)
tags = torch.cat([inner_tags, top_tags, bottom_tags,
initial_tags], dim=0).reshape(-1, 1)
# suffle
idx = torch.randperm(inputs.size(0))
inputs = inputs[idx]
outputs = outputs[idx]
tags = tags[idx]
model = PINN().to(device)
initialize_weights(model)
loss_history = train_model(model, inputs, outputs,
tags, config['sigma'], config['r'], iters=10000)
PDE Loss: 0.0035155173391103745 | Boundary Loss: 810.896728515625
PDE Loss: 0.006064870394766331 | Boundary Loss: 795.9464721679688
PDE Loss: 0.06594930589199066 | Boundary Loss: 664.7308959960938
PDE Loss: 0.41698989272117615 | Boundary Loss: 422.73931884765625
PDE Loss: 7.8050336837768555 | Boundary Loss: 866.4491577148438
PDE Loss: 1.9179744720458984 | Boundary Loss: 206.89183044433594
PDE Loss: 9.82781982421875 | Boundary Loss: 365.9121398925781
PDE Loss: 2.088878631591797 | Boundary Loss: 204.8656463623047
PDE Loss: 2.002225875854492 | Boundary Loss: 204.2563018798828
PDE Loss: 2.081315517425537 | Boundary Loss: 202.72703552246094
PDE Loss: 2.106678009033203 | Boundary Loss: 202.97193908691406
PDE Loss: 2.1070854663848877 | Boundary Loss: 202.94993591308594
PDE Loss: 0.27287378907203674 | Boundary Loss: 458.15289306640625
PDE Loss: 2.115199327468872 | Boundary Loss: 202.66616821289062
PDE Loss: 2.234370708465576 | Boundary Loss: 200.9459228515625
PDE Loss: 2.2679648399353027 | Boundary Loss: 200.65245056152344
PDE Loss: 2.586794853210449 | Boundary Loss: 201.0384979248047
PDE Loss: 2.3889284133911133 | Boundary Loss: 200.08926391601562
PDE Loss: 0.4763438403606415 | Boundary Loss: 443.4243469238281
PDE Loss: 2.4260663986206055 | Boundary Loss: 199.91920471191406
PDE Loss: 1.054399847984314 | Boundary Loss: 398.0643310546875
PDE Loss: 2.458738327026367 | Boundary Loss: 199.6092071533203
PDE Loss: 2.7218222618103027 | Boundary Loss: 197.27378845214844
PDE Loss: 2.7805967330932617 | Boundary Loss: 196.7906036376953
PDE Loss: 3.2406487464904785 | Boundary Loss: 193.63165283203125
PDE Loss: 3.3879852294921875 | Boundary Loss: 192.89488220214844
PDE Loss: 4.552643299102783 | Boundary Loss: 192.00625610351562
PDE Loss: 3.915278911590576 | Boundary Loss: 191.42742919921875
PDE Loss: 3.420907735824585 | Boundary Loss: 287.6389465332031
PDE Loss: 4.133335590362549 | Boundary Loss: 190.960205078125
PDE Loss: 4.531111717224121 | Boundary Loss: 180.77728271484375
PDE Loss: 5.39586877822876 | Boundary Loss: 176.3955841064453
PDE Loss: 5.193947792053223 | Boundary Loss: 176.135009765625
PDE Loss: 5.5137410163879395 | Boundary Loss: 175.02932739257812
PDE Loss: 5.4268622398376465 | Boundary Loss: 173.94369506835938
PDE Loss: 5.459446430206299 | Boundary Loss: 176.2516326904297
PDE Loss: 5.324989318847656 | Boundary Loss: 171.82618713378906
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PDE Loss: 5.016088962554932 | Boundary Loss: 166.06976318359375
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PDE Loss: 3.4155659675598145 | Boundary Loss: 148.0717010498047
PDE Loss: 5.698676586151123 | Boundary Loss: 245.93511962890625
PDE Loss: 4.049826145172119 | Boundary Loss: 136.85787963867188
PDE Loss: 9.512383460998535 | Boundary Loss: 119.10015869140625
PDE Loss: 3.817021131515503 | Boundary Loss: 342.657470703125
PDE Loss: 10.361550331115723 | Boundary Loss: 109.29862976074219
PDE Loss: 11.020111083984375 | Boundary Loss: 97.33594512939453
PDE Loss: 11.487074851989746 | Boundary Loss: 86.61174774169922
PDE Loss: 12.172638893127441 | Boundary Loss: 75.1395263671875
PDE Loss: 12.388835906982422 | Boundary Loss: 69.55958557128906
PDE Loss: 11.615365028381348 | Boundary Loss: 65.48136138916016
PDE Loss: 68.61216735839844 | Boundary Loss: 1537.17529296875
PDE Loss: 8.411243438720703 | Boundary Loss: 73.00325775146484
PDE Loss: 10.539984703063965 | Boundary Loss: 65.77694702148438
PDE Loss: 6.620866298675537 | Boundary Loss: 88.80309295654297
PDE Loss: 6.868873596191406 | Boundary Loss: 64.04460144042969
PDE Loss: 8.48397445678711 | Boundary Loss: 56.27490997314453
PDE Loss: 1.4595659971237183 | Boundary Loss: 134.54092407226562
PDE Loss: 9.469560623168945 | Boundary Loss: 54.0434684753418
PDE Loss: 15.806249618530273 | Boundary Loss: 46.432281494140625
PDE Loss: 13.284977912902832 | Boundary Loss: 47.572731018066406
PDE Loss: 13.461488723754883 | Boundary Loss: 45.720680236816406
PDE Loss: 13.660995483398438 | Boundary Loss: 43.74755859375
PDE Loss: 12.984204292297363 | Boundary Loss: 41.66096115112305
PDE Loss: 11.565495491027832 | Boundary Loss: 38.779483795166016
PDE Loss: 10.504505157470703 | Boundary Loss: 37.983707427978516
PDE Loss: 12.613459587097168 | Boundary Loss: 31.65300750732422
PDE Loss: 12.370379447937012 | Boundary Loss: 28.44278907775879
PDE Loss: 11.791336059570312 | Boundary Loss: 27.074148178100586
PDE Loss: 11.87125015258789 | Boundary Loss: 25.59307098388672
PDE Loss: 10.164714813232422 | Boundary Loss: 25.99994659423828
PDE Loss: 9.717982292175293 | Boundary Loss: 25.041414260864258
PDE Loss: 8.85610294342041 | Boundary Loss: 24.350078582763672
PDE Loss: 8.00014877319336 | Boundary Loss: 22.693574905395508
PDE Loss: 8.786383628845215 | Boundary Loss: 19.9447021484375
PDE Loss: 9.12101936340332 | Boundary Loss: 17.471715927124023
PDE Loss: 17.0146484375 | Boundary Loss: 16.07123565673828
PDE Loss: 10.738739013671875 | Boundary Loss: 15.24732494354248
PDE Loss: 11.518293380737305 | Boundary Loss: 13.252223014831543
PDE Loss: 10.98302936553955 | Boundary Loss: 12.461416244506836
PDE Loss: 10.432538986206055 | Boundary Loss: 12.617741584777832
PDE Loss: 9.740677833557129 | Boundary Loss: 12.427865982055664
PDE Loss: 9.04503345489502 | Boundary Loss: 11.003106117248535
PDE Loss: 12.225879669189453 | Boundary Loss: 7.411933422088623
PDE Loss: 9.706644058227539 | Boundary Loss: 7.894304275512695
PDE Loss: 9.045411109924316 | Boundary Loss: 7.847382068634033
PDE Loss: 8.462629318237305 | Boundary Loss: 7.736846446990967
PDE Loss: 8.813063621520996 | Boundary Loss: 6.804384231567383
PDE Loss: 8.726329803466797 | Boundary Loss: 6.442319393157959
PDE Loss: 8.155579566955566 | Boundary Loss: 5.969675540924072
PDE Loss: 7.3621673583984375 | Boundary Loss: 5.6920485496521
PDE Loss: 5.218063831329346 | Boundary Loss: 5.274911880493164
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PDE Loss: 0.0026091428007930517 | Boundary Loss: 0.008730102330446243
PDE Loss: 0.0026394070591777563 | Boundary Loss: 0.008722091093659401
PDE Loss: 0.0026181747671216726 | Boundary Loss: 0.008701690472662449
PDE Loss: 0.002682098187506199 | Boundary Loss: 0.008631289005279541
PDE Loss: 0.0027169196400791407 | Boundary Loss: 0.008566796779632568
PDE Loss: 0.0027696245815604925 | Boundary Loss: 0.008768010884523392
PDE Loss: 0.0027207890525460243 | Boundary Loss: 0.008552498184144497
PDE Loss: 0.002758330199867487 | Boundary Loss: 0.008510786108672619
PDE Loss: 0.002806503791362047 | Boundary Loss: 0.008478686213493347
PDE Loss: 0.0027725461404770613 | Boundary Loss: 0.008479523472487926
PDE Loss: 0.002853146055713296 | Boundary Loss: 0.008382881991565228
PDE Loss: 0.0028343643061816692 | Boundary Loss: 0.008404982276260853
PDE Loss: 0.0028427066281437874 | Boundary Loss: 0.008385430090129375
PDE Loss: 0.0028839113656431437 | Boundary Loss: 0.008335484191775322
PDE Loss: 0.0028789248317480087 | Boundary Loss: 0.008329622447490692
PDE Loss: 0.0029227621853351593 | Boundary Loss: 0.008280958980321884
PDE Loss: 0.0030292198061943054 | Boundary Loss: 0.008178762160241604
PDE Loss: 0.002964289393275976 | Boundary Loss: 0.008235927671194077
PDE Loss: 0.0029676095582544804 | Boundary Loss: 0.008231347426772118
PDE Loss: 0.002729846630245447 | Boundary Loss: 0.009087493643164635
PDE Loss: 0.00293712317943573 | Boundary Loss: 0.00825973879545927
PDE Loss: 0.00295856106095016 | Boundary Loss: 0.008237875998020172
PDE Loss: 0.00294768949970603 | Boundary Loss: 0.008244684897363186
PDE Loss: 0.002975873649120331 | Boundary Loss: 0.008221925236284733
PDE Loss: 0.002957371762022376 | Boundary Loss: 0.00823255255818367
PDE Loss: 0.002976604038849473 | Boundary Loss: 0.008211958222091198
PDE Loss: 0.0029713900294154882 | Boundary Loss: 0.00821413192898035
PDE Loss: 0.002979930955916643 | Boundary Loss: 0.008200648240745068
PDE Loss: 0.0029395022429525852 | Boundary Loss: 0.008238663896918297
PDE Loss: 0.0029781796038150787 | Boundary Loss: 0.008195733651518822
PDE Loss: 0.002954295836389065 | Boundary Loss: 0.008219467476010323
PDE Loss: 0.0029656377155333757 | Boundary Loss: 0.008205538615584373
PDE Loss: 0.0029403152875602245 | Boundary Loss: 0.008227529004216194
PDE Loss: 0.002884704852476716 | Boundary Loss: 0.008280741050839424
PDE Loss: 0.00285732583142817 | Boundary Loss: 0.008298891596496105
PDE Loss: 0.0028758251573890448 | Boundary Loss: 0.008271973580121994
PDE Loss: 0.0028237919323146343 | Boundary Loss: 0.008320581167936325
PDE Loss: 0.0028572878800332546 | Boundary Loss: 0.008284522220492363
PDE Loss: 0.002926014829427004 | Boundary Loss: 0.008337163366377354
PDE Loss: 0.0028636574279516935 | Boundary Loss: 0.008273490704596043
PDE Loss: 0.0028121517971158028 | Boundary Loss: 0.008329719305038452
PDE Loss: 0.002841023728251457 | Boundary Loss: 0.00829231645911932
PDE Loss: 0.002846960909664631 | Boundary Loss: 0.008283047936856747
PDE Loss: 0.002874632366001606 | Boundary Loss: 0.008250555023550987
PDE Loss: 0.002916294150054455 | Boundary Loss: 0.008237834088504314
PDE Loss: 0.002878395142033696 | Boundary Loss: 0.008245273493230343
PDE Loss: 0.0029178885743021965 | Boundary Loss: 0.008201893419027328
PDE Loss: 0.00295225135050714 | Boundary Loss: 0.008164338767528534
PDE Loss: 0.0029686305206269026 | Boundary Loss: 0.008144417777657509
PDE Loss: 0.002955478383228183 | Boundary Loss: 0.008153142407536507
PDE Loss: 0.0029417723417282104 | Boundary Loss: 0.00816456601023674
PDE Loss: 0.0029478094074875116 | Boundary Loss: 0.008155975490808487
PDE Loss: 0.002998099196702242 | Boundary Loss: 0.008104778826236725
PDE Loss: 0.0029714142438024282 | Boundary Loss: 0.008126980625092983
PDE Loss: 0.0030177130829542875 | Boundary Loss: 0.008076310157775879
PDE Loss: 0.0030312612652778625 | Boundary Loss: 0.008062598295509815
PDE Loss: 0.003024705220013857 | Boundary Loss: 0.008066896349191666
PDE Loss: 0.0030064559541642666 | Boundary Loss: 0.008086692541837692
PDE Loss: 0.00301605137065053 | Boundary Loss: 0.008073796518146992
PDE Loss: 0.0030140080489218235 | Boundary Loss: 0.008072813972830772
PDE Loss: 0.003024151548743248 | Boundary Loss: 0.008055690675973892
PDE Loss: 0.0030475440435111523 | Boundary Loss: 0.008019248023629189
PDE Loss: 0.0031594967003911734 | Boundary Loss: 0.007912794128060341
PDE Loss: 0.003097440814599395 | Boundary Loss: 0.007961557246744633
PDE Loss: 0.003162923501804471 | Boundary Loss: 0.007918622344732285
PDE Loss: 0.0031228798907250166 | Boundary Loss: 0.007929573766887188
PDE Loss: 0.0031044669449329376 | Boundary Loss: 0.007957578636705875
PDE Loss: 0.0031161385122686625 | Boundary Loss: 0.00793374888598919
PDE Loss: 0.0030762110836803913 | Boundary Loss: 0.008020798675715923
PDE Loss: 0.003104323288425803 | Boundary Loss: 0.00794190913438797
PDE Loss: 0.0031234200578182936 | Boundary Loss: 0.007920012809336185
PDE Loss: 0.0031270585022866726 | Boundary Loss: 0.00790939386934042
PDE Loss: 0.0031590149737894535 | Boundary Loss: 0.007874931208789349
PDE Loss: 0.0031349314376711845 | Boundary Loss: 0.00789986178278923
PDE Loss: 0.0031475424766540527 | Boundary Loss: 0.007883994840085506
PDE Loss: 0.0031413822434842587 | Boundary Loss: 0.007887606509029865
PDE Loss: 0.003165261121466756 | Boundary Loss: 0.00786056462675333
PDE Loss: 0.003188883187249303 | Boundary Loss: 0.007834674790501595
PDE Loss: 0.0031886694487184286 | Boundary Loss: 0.00783280748873949
PDE Loss: 0.003208325244486332 | Boundary Loss: 0.007810660172253847
PDE Loss: 0.003212772076949477 | Boundary Loss: 0.007804139517247677
PDE Loss: 0.0032482254318892956 | Boundary Loss: 0.007766743190586567
PDE Loss: 0.0032324637286365032 | Boundary Loss: 0.007778964005410671
PDE Loss: 0.0032082549296319485 | Boundary Loss: 0.0077988747507333755
PDE Loss: 0.0032071724999696016 | Boundary Loss: 0.0077970861457288265
PDE Loss: 0.0032147823367267847 | Boundary Loss: 0.0077829244546592236
PDE Loss: 0.0031511716078966856 | Boundary Loss: 0.00784827396273613
PDE Loss: 0.0031819758005440235 | Boundary Loss: 0.00781196728348732
PDE Loss: 0.003150389064103365 | Boundary Loss: 0.007838164456188679
PDE Loss: 0.0031275409273803234 | Boundary Loss: 0.007850143127143383
PDE Loss: 0.003135183360427618 | Boundary Loss: 0.007869874127209187
PDE Loss: 0.0031243115663528442 | Boundary Loss: 0.007847647182643414
PDE Loss: 0.003141273045912385 | Boundary Loss: 0.007830586284399033
PDE Loss: 0.0031327910255640745 | Boundary Loss: 0.007832380943000317
PDE Loss: 0.003140102606266737 | Boundary Loss: 0.0078122494742274284
PDE Loss: 0.0031443042680621147 | Boundary Loss: 0.0077995252795517445
PDE Loss: 0.0031360003631561995 | Boundary Loss: 0.007812127470970154
PDE Loss: 0.00313956500031054 | Boundary Loss: 0.0078019010834395885
PDE Loss: 0.003168203867971897 | Boundary Loss: 0.0077692982740700245
PDE Loss: 0.0031827830243855715 | Boundary Loss: 0.007751606870442629
PDE Loss: 0.0032196608372032642 | Boundary Loss: 0.007716927211731672
PDE Loss: 0.0031970154959708452 | Boundary Loss: 0.007734835613518953
PDE Loss: 0.003234364790841937 | Boundary Loss: 0.007695424370467663
PDE Loss: 0.003251624759286642 | Boundary Loss: 0.0076770600862801075
PDE Loss: 0.0032464927062392235 | Boundary Loss: 0.007681996561586857
PDE Loss: 0.0032523490954190493 | Boundary Loss: 0.007674647029489279
PDE Loss: 0.0032637501135468483 | Boundary Loss: 0.007661424111574888
PDE Loss: 0.003264988772571087 | Boundary Loss: 0.007654118817299604
PDE Loss: 0.0032521644607186317 | Boundary Loss: 0.007658077869564295
PDE Loss: 0.0032504138071089983 | Boundary Loss: 0.007656143978238106
PDE Loss: 0.0032121771946549416 | Boundary Loss: 0.0076917014084756374
PDE Loss: 0.0032563551794737577 | Boundary Loss: 0.007644252385944128
PDE Loss: 0.003286935854703188 | Boundary Loss: 0.007612030021846294
PDE Loss: 0.0033169277012348175 | Boundary Loss: 0.007581047713756561
PDE Loss: 0.0033181686885654926 | Boundary Loss: 0.007579613011330366
---------------------------------------------------------------------------
KeyboardInterrupt Traceback (most recent call last)
Cell In[12], line 3
1 model = PINN().to(device)
2 initialize_weights(model)
----> 3 loss_history = train_model(model, inputs, outputs,
4 tags, config['sigma'], config['r'], iters=10000)
Cell In[10], line 34, in train_model(model, inputs, outputs, tags, sigma, r, iters)
32 loss_history['total_loss'].append(total_loss.item())
33 return total_loss
---> 34 optimizer.step(closure)
35 return loss_history
File ~/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/optim/optimizer.py:385, in Optimizer.profile_hook_step.<locals>.wrapper(*args, **kwargs)
380 else:
381 raise RuntimeError(
382 f"{func} must return None or a tuple of (new_args, new_kwargs), but got {result}."
383 )
--> 385 out = func(*args, **kwargs)
386 self._optimizer_step_code()
388 # call optimizer step post hooks
File ~/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/utils/_contextlib.py:115, in context_decorator.<locals>.decorate_context(*args, **kwargs)
112 @functools.wraps(func)
113 def decorate_context(*args, **kwargs):
114 with ctx_factory():
--> 115 return func(*args, **kwargs)
File ~/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/optim/lbfgs.py:428, in LBFGS.step(self, closure)
425 def obj_func(x, t, d):
426 return self._directional_evaluate(closure, x, t, d)
--> 428 loss, flat_grad, t, ls_func_evals = _strong_wolfe(
429 obj_func, x_init, t, d, loss, flat_grad, gtd)
430 self._add_grad(t, d)
431 opt_cond = flat_grad.abs().max() <= tolerance_grad
File ~/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/optim/lbfgs.py:50, in _strong_wolfe(obj_func, x, t, d, f, g, gtd, c1, c2, tolerance_change, max_ls)
48 g = g.clone(memory_format=torch.contiguous_format)
49 # evaluate objective and gradient using initial step
---> 50 f_new, g_new = obj_func(x, t, d)
51 ls_func_evals = 1
52 gtd_new = g_new.dot(d)
File ~/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/optim/lbfgs.py:426, in LBFGS.step.<locals>.obj_func(x, t, d)
425 def obj_func(x, t, d):
--> 426 return self._directional_evaluate(closure, x, t, d)
File ~/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/optim/lbfgs.py:280, in LBFGS._directional_evaluate(self, closure, x, t, d)
278 def _directional_evaluate(self, closure, x, t, d):
279 self._add_grad(t, d)
--> 280 loss = float(closure())
281 flat_grad = self._gather_flat_grad()
282 self._set_param(x)
File ~/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/utils/_contextlib.py:115, in context_decorator.<locals>.decorate_context(*args, **kwargs)
112 @functools.wraps(func)
113 def decorate_context(*args, **kwargs):
114 with ctx_factory():
--> 115 return func(*args, **kwargs)
Cell In[10], line 14, in train_model.<locals>.closure()
13 def closure():
---> 14 optimizer.zero_grad()
15 total_loss = 0
17 pde_mask = (tags == 0).squeeze()
File ~/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/_compile.py:24, in _disable_dynamo.<locals>.inner(*args, **kwargs)
20 @functools.wraps(fn)
21 def inner(*args, **kwargs):
22 import torch._dynamo
---> 24 return torch._dynamo.disable(fn, recursive)(*args, **kwargs)
File ~/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/_dynamo/decorators.py:46, in disable(fn, recursive)
44 fn = innermost_fn(fn)
45 assert callable(fn)
---> 46 return DisableContext()(fn)
47 return DisableContext()
48 else:
File ~/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/_dynamo/eval_frame.py:441, in _TorchDynamoContext.__call__(self, fn)
438 except TypeError:
439 filename = None
440 if (
--> 441 (filename is None or skipfiles.check(fn))
442 and (
443 getattr(fn, "__name__", "") not in ["_call_impl", "_wrapped_call_impl"]
444 )
445 and filename not in DONT_WRAP_FILES
446 ):
447 # call to a builtin without a frame for us to capture
448 fn = external_utils.wrap_inline(fn)
450 callback = self.callback
File ~/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/_dynamo/skipfiles.py:405, in check(obj, is_inlined_call)
404 def check(obj, is_inlined_call=False):
--> 405 return check_verbose(obj, is_inlined_call).skipped
File ~/Desktop/dev/libs/notebook/.conda/lib/python3.11/site-packages/torch/_dynamo/skipfiles.py:371, in check_verbose(obj, is_inlined_call)
334 code: Optional[types.CodeType]
337 """
338 This is the main entry point to determine whether an object (function) should be inlined or skipped.
339 Let's illustrate the logic with an example:
(...)
367 we don't want to inline the lower level function call (e.g, f3) by default.
368 """
--> 371 def check_verbose(obj, is_inlined_call=False):
372 if isinstance(
373 obj, (UserFunctionVariable, UserMethodVariable, NestedUserFunctionVariable)
374 ):
375 try:
KeyboardInterrupt:
Results#
Let’s see how the neural network performs in pricing the option. We will compare the network’s predictions against the true option price obtained from the Black-Scholes formula, but first let’s look at the loss evolution during training.
Benchmark#
How does it compare with the analytical solution? Below, we present a comparison between the analytical solution and the solution obtained via the neural network. As we know, the analytical solution for a European call option is given by
where
and \(\Phi\) denotes the cumulative distribution function (CDF) of the standard normal distribution. Below, we show the comparison between the analytical solution and the solution obtained with the neural network.
Show code cell source
def bs_price(s, k, r, sigma, t):
d1 = (np.log(s / k) + (r + 0.5 * sigma ** 2) * t) / (sigma * np.sqrt(t))
d2 = d1 - sigma * np.sqrt(t)
return s * norm.cdf(d1) - k * np.exp(-r * t) * norm.cdf(d2)
exact_surface = np.zeros_like(surface)
for i in range(len(s)):
exact_surface[:, i] = bs_price(
s[i], config['strike'], config['r'], config['sigma'], t)
exact_surface = np.flip(exact_surface.T, axis=1)
/var/folders/cp/l432p0ns1t38qmnyr_3l3r000000gn/T/ipykernel_69285/257154943.py:2: RuntimeWarning:
divide by zero encountered in log
/var/folders/cp/l432p0ns1t38qmnyr_3l3r000000gn/T/ipykernel_69285/257154943.py:2: RuntimeWarning:
divide by zero encountered in divide
Show code cell source
error = np.linalg.norm(exact_surface - surface, 2) / \
np.linalg.norm(exact_surface, 2)
print("Error (2-norm): %.2f%%" % (error * 100))
Error (2-norm): 0.07%
We obtain an error of 0.21% against the exact solution without much tweaking! This demonstrates that even for a relatively simple problem such as pricing a European call option, the PINN approach can achieve a decent accuracy. This result is showcases the potential of this technique for more complex financial models, where analytical solutions are not available.